284 ANALYTICAL MECHANICS
where -/ (r) is the total acceleration. The negative sign ii the right-hand member of equation (I) indicates the fac that the acceleration is directed toward the center, whil< the radius vector is measured in the opposite direction Equations (I) and (II) are the differential (Mutations of th motion of a particle in a central field of force.
218. General Properties of Motion in a Central Field.— Integrating equation (II) we get
r2w=A, (III
where h is a constant. The following properties, which ar direct consequences of equation (III), are common to al motions in central fields of force.
(1) The radius vector sweeps over equal areas in equ« intervals of time.
When the radius vector turns through an angle do it sweep over an area equal to $r*rdO; therefore the rate at whicj the area is described equals
1 o de i 0 i, , ,
r- _ -r-u _ ft - constant.
2 dt 2 2
(2) The angular velocity of the particle varies inverscl; as the square of the distance of the particle from the cente of force. This is evident from equation (III).
(3) The linear velocity of the particle varies inversely a the length of the perpendicular which is dropped upon th direction of the velocity from the center of force.
It was shown on page 87 that
— ^COS^
r
\
where v is the linear velocity and <£ the angle which th velocity makes with a line perpendicular to the radius vecto; Let p denote the length of the perpendicular dropped froi